Some thoughts on theory and practice (From various works in progress) Michael L. Connell, Ph.D. 307 MBH University of Utah Salt Lake City, Utah 84112 ****************************************************** ** CONNELL@GSE.UTAH.EDU <-- THIS ADDRESS WILL WORK! ** ****************************************************** Since this is the beginning point for this discussion strand, let me offer up some possible ideas for a starting point. As I see it there are several key themes drawn which might be discussed here. Most notably the themes of mathematical constructivism as practiced by the participating teachers, technology usage in a constructivist classroom, and mathematics reform in light of the NCTM Standards (National Council of Teachers of Mathematics, Commission on Standards for School Mathematics, 1989). I don't expect that everyone will agree with all that is stated here. If not, why not? I am trying to get the game started... the ball is now in your court!! A little theory... Mathematical Constructivism A primary difficulty in describing constructivist classrooms is that the teachers tend to be constructivists themselves. This often means that the brand of constructivism which is found in an individual classroom is often idiosyncratic and shares only the constructivist label when compared with "constructivist" classrooms from another area. At first glance this appears similar to the state of affairs which existed at the time of the now infamous "New Math". Unlike this earlier situation, however, constructivism is united at a foundational level to a much higher degree than this earlier reform effort. In retrospect there was not a single "New Math, but actually multiple "New Maths" (Davis, R. B. 1989 personal comments). For despite seeming differences in methodology and implementation there is a deep family resemblance which underlies mathematical constructivism, due in large part to the influence Piaget's thinking has had upon the field. One of the main concepts which constructivists in mathematics education borrow from Piaget is that knowledge and action are tightly intertwined. As Piaget succinctly puts it, "...it is therefore from action that we need to start..." (Piaget, 1972, p. 20). This positioning of knowledge in action is one of the major factors separating constructivism from other instructional theories. Although from a logical perspective there may be reasons for viewing knowledge in declarative or propositional form as is done in many cognitive and information processing models, there are equally strong reasons for viewing knowledge as originating in activity (Orton, 1988). This view of the child acquiring knowledge of the world by operating upon it better fits the goals of those who wish to explain the origin of mathematical knowledge and to help children construct their own mathematical beliefs. Most constructivists favor the active representation over the propositional representation. As Romberg and Carpenter (1986) have noted, many of the problems currently facing mathematics instruction is due to the confusion between the "record of knowledge" as denoted in texts and finished products and knowledge in action as evidenced by the learner. Ceci and McNeliss (1987) put this dilemma nicely when they observe that the learner is not granted the luxury of looking at a propositional representation of a finished set of mental activities. Rather the learner must labor through the muck and goo of the experience itself until a personally meaningful representation is formed. A little practice... Technology through this constructivist lens The question of what to do with technology in such a constructivist mathematics classroom is highly problematic. To see why, let us consider a few points. Current thinking in the field, as reflected in the NCTM Standards, has brought into question in the minds of many teachers the need for children to memorize algorithms for the mechanical processing of numerical data. Few would argue that this is a positive step forward, yet, there is an acknowledged need to learn to create algorithms for the purpose of setting up exploration problems in a form recognizable by the computer. Exactly how this dilemma is to be resolved, however, is the subject of considerable debate. Technological tools in mathematics at the elementary level have been traditionally perceived as either computer programs which automate specific problem types (i.e., "solution in a can" programs), calculators and spreadsheets, and programming languages - most typically LOGO. With this background it is not surprising that their impact has been more to replace the labors of computation while offering little in terms of focusing on student problem solving. The case of the "solution in a can program", as exemplified by the automation of a particular class of problems, is the most disturbing of these examples presented. This approach not only does away with the need for any student thinking whatsoever, but so trivializes the problem itself as to make its nuances and learning potential nil. Although it is of great benefit to have programs which compute loan amortization, for example, examination of the output of such programs does little to teach the application of compound interest. Not only are the computation and solution paths masked in such programs, but the student has no opportunity to understand what the answer really means - let alone how it was derived. Calculators and spreadsheets, although significantly more promising, likewise suffer from deficiencies when applied to classroom instruction. The primary difficulty of the calculator lies not in it's potential, but in it's implementation. Calculators are typically used, for example, as a means of checking student work originating from drill and practice applications. Calculator problems in traditional classrooms tend to focus upon the application of a predetermined process for a specific problem and often lead to unintended difficulties. Furthermore, over reliance upon them at too early a point often reinforces student beliefs of inability and does not lend itself to the development of a widely developed conceptual underpinnings for the operations which they so speedily and accurately perform. In addition to a widespread perception that they are not appropriate for elementary school children spreadsheets suffer a similar fate. It should be noted, however, that when students already possess the underlying operational concepts spreadsheets are a powerful alternate route into the algebra (Sutherland & Rajano, 1993). They are closer to the paper and pencil methods used by children and with the development of integrated powerful graphing features allow for a broader range of representation possibilities than is present in the standalone calculator alone. The problem is that they remain a black box to the elementary student with only the outcome being visible. The methods of solution which lead to this answer and rationale remains invisible to the elementary student - thus weakening their potential applicability. And now.... its your turn! Hopefully you have found a few things here worthy of your contemplation. Lets get this newsgroup up and running! Hope to see you in Washington, D.C. at this years STATE conference. .